Evolutionary dynamics with game transitions.

The environment has a strong influence on a population's evolutionary dynamics. Driven by both intrinsic and external factors, the environment is subject to continual change in nature. To capture an ever-changing environment, we consider a model of evolutionary dynamics with game transitions, where individuals' behaviors together with the games they play in one time step influence the games to be played next time step. Within this model, we study the evolution of cooperation in structured populations and find a simple rule: weak selection favors cooperation over defection if the ratio of the benefit provided by an altruistic behavior, $b$, to the corresponding cost, $c$, exceeds $k-k'$, where $k$ is the average number of neighbors of an individual and $k'$ captures the effects of the game transitions. Even if cooperation cannot be favored in each individual game, allowing for a transition to a relatively valuable game after mutual cooperation and to a less valuable game after defection can result in a favorable outcome for cooperation. In particular, small variations in different games being played can promote cooperation markedly. Our results suggest that simple game transitions can serve as a mechanism for supporting prosocial behaviors in highly-connected populations.